Photonic signal frequency up and down-conversion using a photonic band gap structure

ABSTRACT

A photonic band gap (PBG) device is provided for frequency up and/or down-converting first and second photonic signals incident on the device to produce a down-converted output photonic signal. When the first and second incident photonic signals have respective first and second frequencies ω 3  and ω 2 , the down-converted photonic signal has a third frequency ω 1 =ω 3 −ω 2 . When the first incident field has a frequency ω 1 , the first up-converted photonic signal has a second frequency ω 3 =ω 1 +ω 2 . The second up-converted photonic signal has a third frequency ω 3 =ω 1 +ω 2 . Thus, the PBG device can be used to generate coherent near- and mid-IR signals by frequency down-converting photonic signals from readily available photonic signal sources, or red, blue, and ultraviolet signals by up-converting the same readily available photonic signal sources. The PBG device includes a layered stack having a plurality of first material layers and a plurality of second material layers. The first and second material layers are arranged such that the PBG device exhibits a photonic band gap structure exhibiting first, second and third transmission band edges respectively corresponding to the first, second, and third frequencies. An interaction of the first and second photonic signals with the arrangement of layers in the metal stack causes a mixing process to generate the both up and down-converted photonic signal at the third frequency.

CROSS-REFERENCE TO RELATED APPLICATION

This application is related to International Application PCT/US98/06378,with an international filing date of Apr. 2, 1998, now pending(incorporated by reference herein in its entirety).

STATEMENT AS TO RIGHTS TO INVENTIONS MADE UNDER FEDERALLY-SPONSOREDRESEARCH AND DEVELOPMENT

This invention was made with Government support under ContractDAAH01-96-P-R010 awarded by the U.S. Army Missile Command. TheGovernment has certain rights in the invention.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to the generation of photonic signals atfrequencies other than the input signal. In particular, it relates tophotonic signal frequency up and down-conversion.

2. Related Art

In recent years, advances in photonic technology have generated a trendtoward the integration of electronic and photonic devices. These devicesoffer an array of advantages over conventional electronic devices. Forexample, they can provide enhanced speed of operation, reduced size,robustness to environmental changes, such as rapid temperaturevariations, and increased lifetime and ability to handle high repetitionrates. These structures can be made of metals, semiconductor materials,ordinary dielectrics, or any combination of these materials.

The intense theoretical and experimental investigations of photonic bandgap (PBG) structures that have occurred during the last few years areevidence of the widely recognized potential that these new materialsoffer. In such materials, electromagnetic field propagation is forbiddenfor a range of frequencies, and allowed for others. The nearly completeabsence of some frequencies in the transmitted spectrum is referred toas a photonic band gap (PBG), in analogy to semiconductor band gaps.This phenomenon is based on the interference of light; for frequenciesinside the band gap, forward- and backward-propagating components cancancel destructively inside the structure, leading to completereflection.

For example, recent advancements in PBG structures have been made in thedevelopment of a photonic band edge nonlinear optical limiter andswitch. See, “Optical Limiting and Switching of Ultrashort Pulses inNonlinear Photonic Band-Gap Materials”, M. Scalora, et al., PhysicalReview Letters 73:1368 (1994) (incorporated by reference herein in itsentirety). Also, advancements in photonic technology have been achievedwith the development of the nonlinear optical diode. See, “The PhotonicBand-Edge Optical Diode”, M. Scalora, et al., Journal of Applied Physics76:2023 (1994), which is incorporated by reference herein in itsentirety. In addition, the physical processes involved in the photonicsignal delay imparted by a uniform PBG structure are described in detailin Scalora, et al., “Ultrashort pulse propagation at the photonic bandedge: large tunable group delay with minimal distortion and loss,” Phys.Rev. E Rapid Comm. 54(2), R1078-R1081 (August 1996), which isincorporated by reference herein in its entirety.

The frequency conversion of coherent light sources, such as lasers, hasbeen investigated for many years, because of the desirability to expandthe ranges of available output wavelengths. Many different processeshave been utilized, including Raman-shifting, harmonic generation, andquasi-phase-matching techniques. Also important are frequency up- anddown-conversion, and the more general issue of obtaining laser radiationat frequencies generally not accessible with a more direct process.

Harmonic generation involves the non-linear interactions between lightand matter using a suitable non-linear material that can generateharmonics at multiples of the pump signal frequency. Conventionalnon-linear materials include potassium di-hydrogen phosphate (KDP),β-barium borate (BBO), lithium triborate (LBO), lithium niobate(LiNbO₃), and the like. However, the utility of these types ofnon-linear crystals for efficient frequency conversion often depends onproper adjustment of parameters such as non-linear coefficients,phase-matching capabilities, walk-off angle, and angular acceptance.

For example, lithium niobate is conventionally used for second harmonic(SH) generation because its nonlinear Ω⁽²⁾ coefficient is larger thanmost other materials. In addition, the effective magnitude of Ω⁽²⁾ canbe enhanced further by a process called polling. Typically, a certainlength of LiNbO₃ material, ordinarily a few millimeters to a fewcentimeters, is subdivided in sections each on the order of a fewmicrons in thickness. Then, a strong, static electric field is appliedto the material such that the direction of the electric field isreversed in each successive, adjacent section. In effect then, the fieldleaves a permanent impression behind, similar to the impression thatvisible light leaves on a photographic plate, which causes the sign ofthe Ω⁽²⁾ to reverse in a predetermined way in each successive sectionthroughout the length of the material. As a consequence of alternatingthe sign of the nonlinear index of refraction, a technique that is alsoreferred to as quasi-phase-matching (QPM), SH generation from a similarlength of material that is not quasi-phase-matched can be orders ofmagnitude smaller than the phase-matched case.

The reason for this kind of material processing can be explained asfollows. For SH generation, a field at twice the original frequency, isgenerated. In addition to its dependence on field strength, the index ofrefraction of any material also depends on frequency. For typical SHup-conversion, the indices of refraction may differ by as much as 10% ormore; this means that the speed of light in the material may differ bythat amount, causing the two waves, the fundamental and the SH, to getout of phase. As it turns out, by modulating the Ω⁽²⁾, the waves tend toremain in phase, which defines the QPM phenomenon, thus yieldingenhanced SH generation. The fact that the Ω⁽²⁾ changes sign does notmean that there is a material discontinuity, i.e., the sign reversaloccurs in the same material. The sign reversal of the nonlinearcoefficient merely implies that a molecular realignment is induced bytreating the material in a way to cause QPM phenomenon.

QPM devices utilized in frequency conversion are typically on the orderof a few millimeters, perhaps 1-2 centimeters (cm) in length or more.What is needed is a device that performs frequency conversion of a lightsource that is compact in size, has sufficient conversion efficiency,and can be manufactured by conventional techniques.

There are many optical wavebands (that is, optical frequency bands orranges) with good propagation characteristics from which to choose forimaging laser radars. These bands include the near-Infra Red (IR)(1.06-3.0 micrometers (μm)), the mid-IR (3.0-5.0 μm), and the far-IR(9-11 μm). The near- and mid-IR wavebands hold potential for manyapplications, including long distance laser ranging and tracking, anddetection of environmental pollutants, such as NO₂ compounds, forexample. Although the near- and mid-IR wavebands exhibit goodpropagation characteristics, there is a lack of efficient, compactoptical sources in these spectral regions.

Therefore there is a need for an efficient and compact source of opticalradiation in the near- and mid-IR wavebands.

SUMMARY OF THE INVENTION

The present invention provides a photonic band gap (PBG) device thatfrequency down-converts first and second photonic signals incident onthe device to produce a down-converted output photonic signal.Alternatively, in the case of frequency up-conversion, one or morephotonic signals may be combined to produce an photonic signal with ahigher frequency. According to the present invention, in the case offrequency down-conversion, when the first and second incident photonicsignals have respective first and second frequencies ω₃ and ω₂, thedown-converted photonic signal has a third frequency ω₁=ω₃−ω₂. We notethat there is no restriction on the actual values of ω₁, ω₂, and ω₃, andthat two photonic signals might be combined to generate a microwavesignal, for example. In addition, there is no restriction on the numberof waves that are combined to generate one or more down-convertedsignals. The same arguments hold in the case of up-conversion. One ormore photonic signal of frequencies ω₁, ω₂, and ω₃ may be combinedinside an appropriately designed PBG device in order to generate a setof higher frequencies. Thus, the PBG device of the present invention canadvantageously be used to generate coherent signals from the near- tomid-IR and microwave ranges, which includes Terahertz photonic signals,by frequency down-converting photonic signals from readily availablephotonic signal sources. By the same token, the PBG device of thepresent invention can advantageously be used to generate coherentsignals from the near-IR to the green, blue, ultraviolet, and beyond, byfrequency up-converting photonic signals from readily available photonicsignal sources. In other words, photonic signals readily available byother means, such as lasing, can be used to conveniently obtain light atlower or higher frequencies (that is, “mixed-up or -down”), such as thenear-IR, mid-IR, and microwave range on one end of the spectrum, and theultraviolet and beyond on the other end of the spectrum, according topresent invention.

The PBG device for nonlinear frequency conversion, which includes up anddown-conversion, includes a plurality of first material layers and aplurality of second material layers. In general, the first and secondmaterial layers receive first and second photonic signals incident uponthe device. The first and second frequencies have respective first andsecond frequencies. The first and second material layers are arrangedsuch that the PBG device has a photonic band gap structure exhibitingfirst and second transmission band edges respectively corresponding tothe first and second frequencies. An interaction of the first and secondphotonic signals with the arrangement of layers causes the frequencymixing process to generate a third photonic signal having a thirdfrequency that is less than the first and second frequencies in acommensurate manner. The generated third frequency may be the sum and/orthe difference between the first and second frequency, and correspondsto a third PBG resonance in the PBG device. If this correspondence ofthe generated frequency with another band edge resonance is found, thenthe generation process can be orders of magnitude more efficientcompared to QPM systems, which represent the current state of the art.

Further features and advantages of the present invention, as well as thestructure and operation of various embodiments of the present invention,are described in detail below with reference to the accompanyingdrawings.

BRIEF DESCRIPTION OF THE FIGURES

The present invention is described with reference to the accompanyingdrawings. In the drawings, like reference numbers indicate identical orfunctionally similar elements. Additionally, the left-most digit(s) of areference number identifies the drawing in which the reference numberfirst appears.

FIG. 1A is a schematic representation of one embodiment of the presentinvention, a quarter-wave frequency conversion device with a uniform PBGstructure.

FIG. 1B is a diagram of the characteristic index of refraction profileof the uniform PBG structure shown in FIG. 1A.

FIG. 2 is a schematic diagram of one embodiment of the presentinvention, a mixed quarter-half-wave PBG device.

FIG. 3 shows a characteristic transmission profile for a PBG device forthird harmonic generation according to the present invention.

FIG. 4 shows group index versus normalized, dimensionless frequencyprofile according to the present invention.

FIG. 5 shows one embodiment of the present invention, a PBG device witha periodicity defect region.

FIG. 6 is a diagram of the characteristic index of refraction profile ofthe PBG structure shown in FIG. 5.

FIG. 7 shows a transmission versus normalized, dimensionless frequencyfor a 20-period, half-quarter-wave stack.

FIG. 8 shows maximum energy output versus index of refraction.

FIG. 9 shows the pump field eigenmode distribution inside a PBGstructure of the present invention, at the instant that the peak of thepulse reaches the PBG structure.

FIG. 10 shows a second-harmonic eigenmode for the case of FIG. 9.

FIG. 11 shows comparison between the SH energy output from the PBG(solid line) and a phase-matched bulk material (dotted line), as afunction of pulse width.

FIG. 12 shows spontaneously generated SH pulses.

FIG. 13 shows SH conversion efficiency versus incident pulse peak fieldstrength.

FIG. 14 is a flowchart illustrating a method of generating frequencyconversion according to the present invention.

FIG. 15 is a block diagram of an embodiment of a system for frequencydown-converting photonic signals.

FIG. 16 is an embodiment of a PBG device from FIG. 15 capable offrequency down-converting photonic signals incident on the PBG device.

FIG. 17A is a transmission function of the PBG device of FIGS. 15 and16. Tuning a plurality of fields (ω₁, ω₂, and ω₃) as depicted in FIG.17A meets phase matching conditions conducive to photonic signaldown-conversion in the present invention.

FIG. 17B is a transmission function of the PBG device of FIGS. 15 and16. Tuning the plurality of fields (ω₁, ω₂, and ω₃) as depicted in FIG.17B, as an alternative to the tuning of the fields in 17B, also meetsphase matching conditions conducive to photonic signal down-conversionin the present invention.

FIG. 18 includes a series of energy function plots of first and secondphotonic pump fields and a down-converted photonic field resulting frompumping the PBG device of FIG. 15 with the first and second photonicpump fields. Each energy function plot is a plot of total integratedenergy vs. time the associated photonic field.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

1. Overview of the Present Invention

The present invention provides a frequency conversion device thatutilizes a photonic band gap (PBG) structure. The enhancement mechanismdemonstrated in these PBG structures in the linear regime leads tofrequency up- (or down-) conversion rates nearly three orders ofmagnitude better than conversion rates achieved with ordinary phasematched materials, or in conventional fiber grating geometry. Thegeometrical properties and the periodicity of the PBG structure can actto significantly modify the density of electromagnetic field modes andphase matching conditions near the band edge, thus facilitating theemission of the second harmonic (SH) signal at a much-enhanced rate.More importantly perhaps, this means that current fabrication issuesthat arise in ordinary quasi-phase-matched structures can be avoidedaltogether by utilizing current technology for deposition ofsemiconductor or dielectric thin films and combinations thereof.

The present invention is described in terms of this example environment.Description in these terms is provided for convenience only. It is notintended that the invention be limited to application in this exampleenvironment. In fact, after reading the following description, it willbecome apparent to a person skilled in the relevant art how to implementthe invention in alternative environments.

2. Nonlinear Interaction of Light with Matter

The nonlinear interaction of light with matter is important forapplications in the field of light generation at frequencies that areusually not accessible by a more direct laser process. At a qualitativelevel, all materials found in nature are nonlinear to some degree. Thismeans that the characteristic properties of ordinary materials, such asthe dielectric susceptibility, change if an applied electromagneticfield intensity is strong enough.

This dependence of the susceptibility (which ultimately is a statementof the index of refraction of the material) on the electric fieldstrength can be exemplified in the following way:Ω=Ω⁽¹⁾+Ω⁽²⁾ E+Ω⁽³⁾ E ²+ . . . Ω^((j)) E ^(j−1) . . . + . . .where j is an integer, Ω⁽¹⁾ is the medium susceptibility for lowincident fields, Ω^((j)) is the jth nonlinear coefficient whosemagnitude decreases rapidly as (j) increases, and E is the appliedfield. Therefore, contributions from the jth term become significant ifthe field strength is gradually increased. Typically, the Ω^((j)) can betwo to four orders of magnitude greater than each successive Ω^((j+1))coefficient, depending on the material. On the other hand, all thecoefficients with odd or even (j) greater than one may vanish, dependingon the characteristics of the material, such as symmetry properties, atthe molecular level. For example, all the even coefficients vanish ifthe molecule has a geometrical center of symmetry, as in a gas.

Because of the nonlinear contributions to the dielectric susceptibility,the application of a strong external optical field at frequency ω, forexample, is capable of generating light at frequency 2ω, 3ω, 4ω, and soon. By the same token, if two strong fields of different frequencies ω₁and ω₂ are applied to the nonlinear material, light at frequencies(ω₁+ω₂) and (ω₁−ω₂) (i.e., sum and difference frequencies) can also begenerated in addition to the individual harmonics. For example, a Ω⁽²⁾medium, which means that the first order nonlinear coefficient dominatesthe dynamics, is capable of SH generation, and sum and differencefrequency conversion; a Ω⁽³⁾ medium is capable of third harmonicgeneration, and so on.

For example, a type of nonlinear frequency conversion that is typicallysought in nonlinear media is SH generation. However, the presentdescription is also applicable for nonlinear frequency conversion tohigher or lower frequencies, such as third harmonic generation, and soon.

Conventional nonlinear materials used for frequency conversionprocesses, such as LiNbO₃, are processed in such a way that thenonlinear contribution to the index of refraction alternates sign everyfew tens of microns. However, the linear index of refraction of theLiNbO₃ host material is not modified in any way (i.e., it is spatiallyuniform).

The method of forming a device designed to perform frequency conversion,according to the present invention, is completely different: a spatialmodulation is imparted to the linear part of the refractive index. Inother words, the linear index of refraction of the structure alternatesbetween a high and a low value, which can happen only if the materialsare different. This is accomplished by alternating at least twomaterials, such as GaAs (Gallium Arsenide) and AlAs (Aluminum Arsenide),whose indices of refraction are approximately 3.4 and 2.9 respectively,resulting in a structure between 5 and a few tens of microns in length.The consequence of alternating different materials with differentrefractive indices as indicated above is the creation of a photonic bandgap (PBG) structure.

The physical processes that are exploited in the present invention aredifferent from conventional frequency conversion techniques in thatphotonic band edge effects are utilized. Photonic band edge effectscause strong overlap of the pump and SH signals, significant reductionof the respective propagation velocities, and therefore, increasedinteraction times, and exact phase matching conditions which do notordinarily exist in nature. As described below, some of the advantagesof the present invention include: (1) the structure can be 100 to 1000time shorter than typical QPM structures, with comparable conversionefficiencies; (2) ordinary semiconductor materials can be used informing the PBG structure, leading to a reduction of production costs;and (3) the PBG device is compatible with integrated circuitenvironments due to its size and composition.

3. Frequency Conversion using a PBG Structure

In one dimension, a photonic band gap structure comprises a plurality oflayers, as shown in FIG. 1A, where the plurality of layers alternatesbetween a low and a high index of refraction. PBG structure 102comprises a stack of alternating layers 108 and 110 of refractivematerials having predetermined indices of refraction n₁ and n₂ (for lowincident pump powers), and predetermined thicknesses a and b,respectively. In particular, the first type of layer 108 can be chosensuch that it is a high index layer n₁. The second type of layer 110 canbe chosen to be a low index layer n₂. The widths of the layers can bechosen such that they are both a fraction of the size of the incidentpump wavelength. We note that it is possible to combine more than twotypes of layers to form a photonic band gap structure in a known manner.For example, a third type of layer of index n₃ can be added to the stackwhile the pass bands and band gaps still remain visible. For simplicity,however, all the example that we will show consist of only two types oflayers.

For example, first layer 108 can be designed to have a thickness (a)corresponding to the wavelength of an incoming photonic signal (λ),determined by the equation a=λ/4n₁. Similarly, second layer 110 can havean index of refraction n₂, and a thickness (b), where b=λ/4n₂. Thispattern can be repeated for N periods 122, where a period is equal toone set of alternating layers 112. This type of structure is alsoreferred to as a quarter-wave structure. As would be apparent to one ofordinary skill in the art based on the present description, otherarrangements of alternating layers can also be made, depending on theparticular frequency conversion application, and structures that are notnecessarily periodic are also envisioned. This means that thethicknesses of one or several layers inside the stack are notnecessarily equal or commensurate with the other layers inside thestack. For example, one layer or more layers might be chosen to be ½, ¾,or any other fraction which might be helpful in the design process.Adjusting layer width causes a shift of the location of the band gap toa different frequency. This property is a beneficial and useful one,which as we will see below is the key to the flexibility andfunctionality of the device when the options of input and output laserfrequencies are being considered.

FIG. 1B is a diagram of a characteristic index of refraction square-wavepattern profile of PBG structure 102 for N periods. Diagram 150 plotsthe index of refraction (n) 152 of a uniform PBG structure as a functionof distance (z) 154, which is limited by the number of periods 156 inthe device. Diagram 150 illustrates the periodic nature of the abruptrefractive index changes occurring in the material.

In general, large index modulation PBG structures (that is, the indexdifference between adjacent layers can be of order unity) are notsusceptible to nonlinear index changes because index variations are asmall perturbation on the linear index modulation depth. As describedbelow, for ultrashort pulses tuned near the photonic band edge, a choiceof materials with suitable indices of refraction, thicknesses, andperiodicity can lead to low group velocities, enhanced field intensity,exact phase matching conditions, and conversion efficiencies nearlythree orders of magnitude larger than conventional QPM, bulkup-conversion rates. Conversion efficiencies greater than 10⁻³ (that is,at least one part in a thousand of the original beam's energy isconverted to the second frequency) can be achieved for structures only afew micrometers in length, with a single pump pass, and at realisticpump intensities. Plane-wave conversion rates can be approximated byutilizing pulses whose frequency bandwidth is smaller than thetransmission resonance bandwidth, such as for pump signals only a fewpicoseconds (ps) in duration.

A preferred embodiment of the present invention is shown in FIG. 2. PBGstructure 200 is formed in such a way that a single period comprises twobasic layers: a quarter-wave layer 202 and a half-wave layer 204, toform a periodic, mixed quarter-half-wave structure. This particularchoice causes the first and second order band edges to be approximatelya factor of two apart from each other, as indicated in FIG. 4, describedin detail below. Then, both the pump and SH fields are tuned to theirrespective photonic band edges. This coincidence of the band edges leadsto strong overlap of the fields, significant reduction of the wavevelocities by several orders of magnitude below the speed of light ineither medium, increased interaction times, and exact phase matching.See, e.g., “Pulsed second harmonic generation in one-dimensional,periodic structures”, Phys. Rev. A, October 1997, by Scalora et al., and“Dispersive properties of finite, one-dimensional photonic band gapstructures: applications to nonlinear quadratic interactions”, Phys.Rev. E. October 1999, (both incorporated by reference herein in itsentirety). These factors result in an increase in the SH energy outputthat significantly exceeds conventional QPM, current state of the artdevices.

The types of structures discussed above result in a PBG structure inwhich a range of frequencies about some reference frequency cannotpropagate inside a PBG device. On the other hand, the structure may betransparent to other frequencies away from the band gap. For example, arepresentative photonic band gap structure is shown in FIG. 3, whichshows a characteristic transmission profile for structure 301. At higherfrequencies, higher order gaps may also appear, to create a series ofgaps, which may be useful for either up or down-conversion processes.Under ordinary circumstances, higher order gaps are ignored. In FIG. 3,both the first order band gap 302 and second order band gap 304 aredepicted. Typically, a uniform PBG structure, such as that shown in FIG.1A, exhibits an infinite number of photonic band gaps and band edges. InFIG. 3, transmission profile 306 is obtained by plotting the opticaltransmission 308 as a function of normalized frequency (Ω) 310, whereΩ=ω/ω₀. The maximum possible transmission is 1, which corresponds to100%. Therefore, it is the absence of those frequencies from thetransmitted spectrum that gives rise to the name “band gap”, in analogyto the electronic band gap of semiconductors where electrons having aspecific range of energies cannot propagate inside a semiconductorcrystal.

At frequencies outside the photonic band gap, the properties of thestructures are such that a series of transmission resonances areobtained. The number of such resonances is equal to the number ofperiods that make up the structure. The bandwidth of said resonances isa sensitive function of the total number of periods, the indices n₁ andn₂, and their difference δn=|n₂−n₁|, also known as index modulationdepth.

In regard to SH generation, a PBG structure can be formed wherenonlinear gain, or the production of SH signal, is maximized. Using thecalculations described in detail below, the equations that describe thepropagation of electromagnetic waves in PBG structures can be solved.The results of the calculations show that if light signal interacts witha nonlinear Ω⁽²⁾ medium to produce a SH signal, then the SH energyoutput from the PBG structure is in this case approximately three ordersof magnitude greater than the energy output from a bulk, phase matchednonlinear medium of approximately the same length. Conditions andstructure design can be improved to further enhance conversion rates.

One embodiment of the present invention is a PBG structure thatcomprises 20 periods (or 40 layers) of alternating layers of GaAs andAlAs. Alternatively, the PBG structure can also comprise different setsof materials, for example, air and GaAs, glass and AlAs, AIN and GaN,AIN and SiO₂, or a combination of other dielectric materials, as well aswith materials that would not conventionally be considered as nonlinearmaterials. In addition, the PBG structure may also be created in anoptical fiber, in the form of a fiber grating. This illustrates thatthis frequency conversion capability is not specific to any onematerial, and that some flexibility exists according to the specificneeds of a particular application. Accordingly, the structure of thepresent invention should not be limited solely to the embodimentsdescribed herein.

According to the present invention, a pulse of light of about onepicosecond or more (more here means that is possible to think of anincident plane wave) in duration can be tuned to the frequencycorresponding to the maximum of the first transmission resonance awayfrom the low frequency band edge. This is shown schematically in FIG. 4,which plots group index as a function of normalized frequency. The totalenergy of a signal produced at twice the frequency of a pump (i.e., atthe SH frequency) is about 1000 times greater than the energy output ofa QPM device of similar nonlinear properties and dimensions, but thatdoes not exhibit a photonic band gap structure. Accordingly, the signalgenerated at the second harmonic frequency is tuned to the secondtransmission resonance of the low frequency band edge of the secondorder gap, as shown in FIG. 4.

According to the present invention, if the enhancement of any otherfrequency is desired, for example, the difference or the sum of theincident frequencies, it will be apparent to one of skill in the artbased on the present description to devise a PBG structure such thatboth the pump and the desired frequency are both tuned to a photonicband transmission resonance. If higher conversion efficiencies aresought, the calculations explained in detail below indicate that suchconversion efficiency increases can be accomplished with only modestincreases in the number of periods that comprise the structure. Thereason for this is that the conversion efficiency in a typical PBGstructure is sensitive to the length of the structure. For example, if Nis the number of periods comprising the device, then the energy outputis approximately proportional to the N⁶. In contrast, for a bulk, QPMmaterial, the conversion efficiency is proportional only to N². As wouldbe apparent to one of skill in the art based on the present description,an optimization procedure can then be employed to produce the idealparameters for the up- or down-conversion process for a particularapplication.

For example, in the case of third harmonic generation, a PBG structurecomprises a quarter-wave periodic structure with a “defect” layer onehalf wavelength thick at the center of the structure. This embodiment isshown schematically in FIG. 5. Device 502 comprises at least two stacks(or regions) 504 and 506 of alternating layers of refractive materialssimilar to those described above in connection with FIG. 1A. In thecenter of device 502, a periodicity defect region 508 is interposed (orplaced) between stacks 504 and 506, with each stack having an equalnumber of alternating layers of refractive material. Defect region 508is also a refractive material that can have an index of refraction (n)that is equivalent to either n₁ or n₂, and with the same Ω⁽²⁾ nonlinearcoefficient. For example, if individual layer thicknesses in the uniformstacks 504 and 506 are taken to be one quarter-wavelength long, then thethickness of periodicity defect region 508 can be one half or onewavelength in thickness. However, other thicknesses for periodicitydefect region 508 can also be utilized. The term “defect”, in thiscontext, simply means a break in the periodicity of the structure.

This defect layer breaks the periodicity in such a way that it generatesa transmission resonance in the middle of every gap, as shown in FIG. 6.Here, the distance between the center of the first and second order gapis exactly a factor of three. Therefore, tuning the pump signal to thecenter of the first order gap will enhance the generation of light atthe third harmonic. For example, using a pump signal wavelength ofapproximately 1550 nm, such as found in conventional communicationslaser diodes, a third harmonic signal will be output from the PBG deviceat a wavelength of approximately 516 nanometers (nm). Therefore, byselecting the proper set of parameters, such as material type, materialparameters, and the exact geometrical properties of the materials (i.e.,layer thickness), a person of skill in the art can arrive at a devicewith the desired properties.

Another embodiment of the present invention is a PBG device comprising aplurality of periodicity “defects.” In other words, several defects ofvarying thicknesses can be placed in a PBG device. The placement ofthese multiple defects between stacks of alternating layers forms ana-periodic structure that also exhibits a photonic band gap structure.This a-periodic structure can be utilized to perform any of thefrequency conversion techniques described herein, as would be apparentto one of skill in the art based on the present description.

According to the present invention, conversion efficiencies can be evenhigher for structures with an increased number of periods. For example,by increasing the structure length by 50% (from 20 to 30 periods), theenergy output can increase by a factor of 5. However, it should be notedthat: (1) the transmission resonance bandwidth decreases as 1/N², whereN is the number of periods, so that the pulse duration needs to beincreased in order to ensure large pump enhancement inside thestructure; and (2) a material breakdown may occur because of excessiveelectric-field buildup, or enhancement, inside the PBG structure.

Consideration of nonlinear effects highlight even more dramaticdifferences between the PBG structures of the present invention andconventional nonlinear materials used for frequency conversion. Typicalnonlinear index changes in GaAs or AlAs layers can be of orderδn_(NL)≈10⁻³. This implies that nonlinear index shifts can be largerthan the linear index modulation depth found in optical fibers.Consequently, the location of the gap on the frequency axis can shiftdramatically to higher or lower frequencies, and its bandwidth canincrease or decrease significantly, depending on the sign of thenonlinearity.

The frequency bandwidth of an ultrashort pulse of only a few hundredoptical cycles in duration (i.e., in the femto-second regime) can besmaller (depending on the wavelength) than the bandwidth of the PBG'sfirst transmission resonance peak, where the group velocity is aminimum. Here, ultrashort pulse propagation can be nondispersive, i.e.,the pulse propagates without breaking up or distortions. In addition,the nonlinear index change remains orders of magnitude smaller than theindex modulation depth, which for PBG structures can be of order unityor larger. Thus, gap and transmission resonance bandwidths, and theirlocations are only marginally altered, although under the rightconditions changes may be sufficient for the onset of optical limitingand switching, optical diode behavior, and strong pulse reshaping.

The stability of the band structure in the frequency domain is alsoimportant in parametric optical up- and down-conversion, and harmonicgeneration. This result highlights the fact that a new generation ofcompact and efficient high gain optical amplifiers and opticalparametric oscillators based on photonic band-edge effects can beachieved according to the present invention.

The enhancement of gain in these PBG structures is understood byrecalling that the density of accessible field modes in the vicinity ofdielectric boundaries is modified by the boundary. This means that if alinear or nonlinear gain medium is introduced with in a PBG structure,the stimulated and spontaneous emission rates are modified according toFermi's golden rule (see below). In QPM structures, a minimization ofthe phase difference between the waves is desirable in order to avoid aphase mismatch in the continuous wave case. For QPM devices, thisminimization of phase difference is typically achieved by poling theactive material, which is uniform in its composition and contains nolinear index discontinuities. Accordingly, the nonlinear coefficientonly alternates sign in the longitudinal direction every few tens ofmicrometers (μm).

For the PBG structures of the present invention, the unusually strongconfinement of both the pump and the SH signal that occurs near thephotonic band edges is relied upon. Where the density of electromagneticfield modes is large and the group velocity is low, the field amplitudemay be enhanced over bulk values by one order of magnitude or more, andstrong pump and SH mode overlap occurs. In this regime, the material isnot poled in the usual manner; it is the geometrical properties of thestructure that cause strong mode overlap, co-propagation, largerinteraction times, and exact phase matching, the combination of which isultimately responsible for the enhanced gain of these PBG structures.

The PBG structures discussed above can be manufactured by conventionaltechniques. Other suitable modifications and adaptations of the varietyof reaction conditions and parameters normally encountered in preparingphotonic and semiconductor devices will be apparent to those skilled inthe art, without departing form the spirit and scope of the invention.

As discussed above, the invention can be implemented in group III-V orII-VI material systems, as well as with dielectric materials. Forpurposes of explanation, the above examples are described in GaAs/AlAsmaterial systems, but it will be understood by those skilled in the artthat the invention described herein can also be implemented with otherIII-V or III-VI systems.

Further, background material concerning semiconductor solid-statephysics may be found in a number of references including two books by S.M. Sze, titled: Physics of Semiconductor Devices, John Wiley and Sons,Inc., New York (1981), and Semiconductor Devices, Physics andTechnology, John Wiley and Sons, Inc., New York (1985), both of whichare incorporated herein by reference. Those skilled in the art canreadily manufacture the layered devices disclosed according toconventional processing techniques without undue experimentation.

4. Example Applications

The PBG structures of the present invention can be utilized to perform avariety of frequency conversion techniques. As described above, a mixedquarter-half-wave structure can be utilized to perform SH generation ofa variety of coherent light sources, including tunable solid statelasers, gas lasers and semiconductor diode lasers. For example, a PBGstructure can be placed at the output facet of a conventional AlGaAsdiode laser that emits a laser beam at a wavelength of approximately 810nm. Diode lasers of various output wavelengths are commerciallyavailable from a number of commercial vendors, including Spectra DiodeLabs, Inc. and Coherent Inc., both of California. By choosing the properset of alternating layer materials, by selecting an appropriate set oflayer thicknesses, and by choosing an appropriate number of periods forthe PBG device, an output at approximately 405 nm can be achieved fromthe PBG device. This type of device would be outputting “blue” laseremission, which is extremely valuable for communications and opticalstorage applications, for example. In addition, because of the compactsize and angular independence of the PBG device (as opposed toconventional non-linear materials such as potassium dihydrogen phosphate(KDP), β-barium borate (BBO), lithium triborate (LBO), which areextremely dependent upon angular alignment), SH generation opticalcavity arrangements (e.g., external cavity and intra-cavity designs)would be very straightforward to design. Typical optical layouts forharmonic generation are well known. See e.g., W. Koechner, “Solid-StateLaser Engineering,” Springer-Verlag, 2^(nd) Ed. (1988), especiallyChapter 10, which is incorporated by reference herein. Knownanti-reflection coatings can also be utilized to reduce spuriousreflections, as would be apparent to one of skill in the art.

In addition, the PBG structures of the present invention can also beutilized in parametric oscillation techniques where, for example, outputwavelengths greater than the pump pulse wavelengths can be generated.Based on the known methods of optical parametric oscillation, such asthose described in the Koechner reference, it would be apparent to oneof skill in the art to design a parametric device utilizing the PBGstructure of the present invention to achieve frequency conversion atlower frequencies (i.e., longer wavelengths).

Further, optical fiber gratings can be designed similar to the types ofPBG structures described above. Optical fiber gratings are also periodicstructures. The index of refraction for a fiber grating can achieve anindex modulation depth (i.e., a high and low value) similar to that ofhigh index contrast semiconductor structures. However, fiber gratingsare structures with a smaller index discontinuity than that associatedwith a semiconductor PBG structure: for a fiber grating an indexmodulation along its core is typically on the order of δn=10⁻³ to 10⁻⁴,as opposed to a semiconductor PBG structure with an index modulationapproaching unity. Since the bandwidth of transmission resonances andband gaps are proportional to δn (the index modulation depth), fibergrating frequency conversion devices are preferred for use with opticalpulses of longer (i.e., nanosecond) duration in order to preserve theirshape.

A fiber grating can be created on an optical fiber by well-knownfabrication techniques. For example, see the fiber grating applicationsand fabrication techniques described in “Continuously tunablesingle-mode erbium fiber laser,” by G. Ball and W. Morey, OpticsLetters, Vol. 17, No. 6, p.420 (1992) and “Spatially-multiplexedfiber-optic bragg grating strain and temperature-sensor system based oninterferometric wavelength shift,” by Y. Rao, et al., ElectronicsLetters, Vol. 31, No. 12, p. 1009 (1995), which are both incorporated byreference in their entirety.

For example, fiber grating fabrication can be accomplished by placing anoptical “mask” over a photo-sensitive fiber core and then byilluminating the mask-fiber assembly with a high intensity ultravioletlight beam, such as an Excimer laser. The resulting grating, referred toas a fiber grating, displays much the same properties of a high indexcontrast semiconductor PBG structure, especially with respect to bandgaps and transmission resonances. In addition, a mask can be designed tocreate a grating that imparts either a band-edge effect or atransmission resonance similar to the one shown above in FIG. 5. Basedon the present description, it would be apparent to one of skill in theart to design a fiber grating capable of frequency conversion. Forexample, a fiber grating device designed according to the parametersdiscussed above can be coupled to the output of a laser diode to producea compact source capable of output emissions in the blue wavelengthrange.

5. Model

1. Equations

According to the present invention, a model can be utilized to allow oneof ordinary skill in the art to design a PBG structure to performoptical frequency conversion for a desired application. For example,shown here is an analysis describing the dynamics associated withultrashort pulses (about 1 μs or less) in one-dimensional systems. Thismodel extends the analysis of SH generation and enhancement toarbitrarily deep PBG gratings in the pulsed regime by directlyintegrating Maxwell's equations in the time domain.

Consider the following simple one-dimensional system similar to device200 shown in FIG. 2. The example device comprises 40 dielectric layers(20 periods in all, roughly 12 μm thick for a reference wavelength of 1μm), and the index of refraction alternates between a high and a lowvalue, n_(2=1.42857) and n₁=1. A small value of Ω⁽²⁾≈0.1 pm/V (roughly3×10⁻⁹ cm/statvolt in Gaussian units) is chosen and it is assumed thatthe nonlinear material is distributed uniformly throughout the PBGstructure. Then, for a reference wavelength λ₀, the layers havethicknesses a=λ₀/(4n₁) and b=λ₀/(2n₂), respectively. This forms a mixedhalf-quarter-wave stack for wavelength λ₀. A range of frequencies isreflected, as shown in FIG. 7, where the transmission coefficient forthis structure is plotted as a function of the scaled frequency Ω=ω/ω₀,where ω₀=2 πc/λ₀. FIG. 7 indicates that this choice of parameters causesthe location of the second-order gap to be removed from the first-ordergap by approximately a factor of 2. For an ordinary quarter-wavestructure, such as the device shown in FIG. 1, a factor of 3 separatesthe first- and second-order band edges. Utilizing these two edges ismore suitable for third-harmonic generation.

The equations of motion can be derived beginning with Maxwell's equationfor the total field, in Gaussian units, and can be written as:$\begin{matrix}{{{\frac{\partial^{2}}{\partial{\mathcal{z}}^{2}}{E\left( {{\mathcal{z}},t} \right)}} - {\frac{n^{2}}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}{E\left( {{\mathcal{z}},t} \right)}}} = {\frac{4\pi}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}{P_{NL}.}}} & (1)\end{matrix}$

Here, P_(NL) is the total nonlinear polarization. Without loss ofgenerality, the fields can arbitrarily and conveniently be decomposed asfollows:E(z,t)=ε_(w)(z,t)e ^(i(kz−ωt)) +c.c.+ε _(2ω)(z,t)e ^(2i(kz−ωt))+c.c.,  (2)P _(NL)(z,t)=ρ_(ω)(z,t)e ^(i(kz−ωt)) +c.c.+ρ _(2ω)(z,t)e ^(2i(kz−ωt))+c.c.  (3)

This decomposition highlights the fundamental and second-harmonicangular frequencies. The nonlinear polarization can be expanded inpowers of the electromagnetic field strength as follows: $\begin{matrix}{{P_{NL}\left( {{\mathcal{z}},t} \right)} = {{\chi^{(2)}{E^{2}\left( {{\mathcal{z}},t} \right)}} = {{2\chi^{(2)}{ɛ_{\omega}^{*}\left( {{\mathcal{z}},t} \right)}{ɛ_{2\omega}\left( {{\mathcal{z}},t} \right)}{\mathbb{e}}^{2{{\mathbb{i}}{({{k\mathcal{z}} - {\omega\quad t}})}}}} + {c.c} + {\chi^{(2)}{ɛ_{\omega}^{2}\left( {{\mathcal{z}},t} \right)}{\mathbb{e}}^{2{{\mathbb{i}}{({{k\mathcal{z}} - {\omega\quad t}})}}}} + {c.c.}}}} & (4)\end{matrix}$

While one can assume an initial left- or right-propagating pump pulse,the SH signal is initially zero everywhere. The direction of propagationof the spontaneously generated SH field and the exact nature of thequasi-standing wave inside the structure are dynamically determined by(a) the nature of the initial and boundary conditions, (b)pump-frequency tuning with respect to the band edge, and (c) thedistribution of nonlinear dipoles inside the structure. This nonlineardipole distribution can significantly affect the results. SH generationis a phase-sensitive process. The field and its phase at any pointinside the structure are a superposition of all fields originatingeverywhere else inside the structure. Thus, the phase is importantelement that should be included in the integration of the equations ofmotion. However, dipole distribution is important to the extent that itis modified in the layers where the fields happen to be localized. Forexample, near the low-frequency band edges, the fields are localized inthe high-index layers. Modifying the nonlinear medium distribution inthe low-index layers will have little effect, although some mode overlapbetween layers always occurs.

For this model, ultrashort incident pulses propagating in the presenceof large index discontinuities are considered. Therefore, allsecond-order spatial derivatives should be retained in order to properlyinclude boundary conditions. However, it can be assumed that pulseenvelopes have a duration that is always much greater than the opticalcycle, thus allowing the application of the slowly varying envelopeapproximation in time (SVEAT) only. For a general description of SVEAT,see Scalora, M., et al., Phys. Rev. Lett. 73:1368 (1994) and referencestherein, which is incorporated by reference herein in its entirety. Theequations of motion for the fundamental and the second-harmonic fieldscan be derived as follows. First, substituting Eq. (2) into Eq. (1)yields: $\begin{matrix}{{{\frac{\partial^{2}ɛ_{\omega}}{\partial{\mathcal{z}}^{2}} + {2{ik}\frac{\partial ɛ_{\omega}}{\partial{\mathcal{z}}}} - {k^{2}ɛ_{\omega}} - {\frac{n_{\omega}^{2}}{c^{2}}\frac{\partial^{2}ɛ_{\omega}}{\partial t^{2}}} + {\frac{2i\quad\omega\quad n^{2}}{c^{2}}\frac{\partial ɛ_{\omega}}{\partial t}} + {\frac{\omega\quad 2}{c^{2}}n_{\omega}^{2}ɛ_{\omega}}} = {\frac{4\pi}{c^{2}}\left( {{\frac{\partial^{2}}{\partial t^{2}}\rho_{\omega}} - {2i\quad\omega} - {\frac{\partial}{\partial t}\rho_{\omega}} - {\omega^{2}\rho_{\omega}}} \right)}},} & (5) \\{{\frac{\partial^{2}ɛ_{2\omega}}{\partial{\mathcal{z}}^{2}} + {4\quad i\quad k\frac{\partial ɛ_{2\omega}}{\partial{\mathcal{z}}}} - {4k^{2}ɛ_{\omega}} - {\frac{n_{2\omega}^{2}}{c^{2}}\frac{\partial^{2}ɛ_{2\omega}}{\partial t^{2}}} + {\frac{{4i\quad\omega}\quad}{c^{2}}n_{2\omega}^{2}\frac{\partial ɛ_{2\omega}}{\partial t}} + {4\frac{{\omega\quad}^{2}}{c^{2}}n_{\omega}^{2}ɛ_{2\omega}}} = {\frac{4\pi}{c^{2}}\left( {{\frac{\partial^{2}}{\partial t^{2}}\rho_{2\omega}} - {4i\quad\omega\frac{\partial}{\partial t}\rho_{2\omega}} - {4\omega^{2}\rho_{2\omega}}} \right)}} & (6)\end{matrix}$where k=ω/c, and the SVEAT is made. This choice of wave vector is simplyan initial condition consistent with a pump field of frequency ωinitially propagating in free space, located away from any structure.Any phase modulation effects that ensue from propagation (i.e.,reflections and nonlinear interactions) are accounted for in thedynamics of the field envelopes. The inclusion of all second-orderspatial derivatives in the equations of motion means that reflectionsare accounted for to all orders, without any approximations. Therefore,assuming that pulses never become so short as to violate SVEAT (usuallythis means a few tens of optical cycles if propagation distances are onthe order of pulse width), neglecting all but the lowest order temporalcontributions to the dynamics, and using the nonlinear polarizationexpansions of Eqs. (4), Eqs. (5) and (6) become: $\begin{matrix}{{{n_{\Omega}^{2}{ɛ_{\Omega}\left( {\xi,\tau} \right)}} = {{\frac{i}{4\pi\quad\Omega}\frac{\partial^{2}ɛ_{\Omega}}{\partial\xi^{2}}} - \frac{\partial ɛ_{\Omega}}{\partial\xi} + {i\quad{\pi\left( {n_{\Omega}^{2} - 1} \right)}\Omega_{ɛ\Omega}} + {{i8\pi}^{2}{\Omega\chi}^{(2)}ɛ_{\Omega}^{*}ɛ_{2\Omega}}}},} & (7) \\{{n_{2\Omega}^{2}{ɛ_{2\Omega}\left( {\xi,\tau} \right)}} = {{\frac{i}{8\pi\quad\Omega}\frac{\partial^{2}ɛ_{2\Omega}}{\partial\xi^{2}}} - \frac{\partial ɛ_{2\Omega}}{\partial\xi} + {i\quad{\pi\left( {n_{2\Omega}^{2} - 1} \right)}2{\Omega ɛ}_{2\Omega}} + {{i8\pi}^{2}{\Omega\chi}^{(2)}{ɛ_{\Omega}^{2}.}}}} & (8)\end{matrix}$

Here ξ=zλ₀, and τ=ct/λ₀. Equation (8) describes the rate of change ofthe SH field, whereas equation (7) describes the pump (or fundamental)signal. The spatial coordinate z has been conveniently scaled in unitsof λ₀; the time is then expressed in units of the corresponding opticalperiod. Thus, by knowing the indices of refraction for the layers of thePBG device, the pump signal frequency and bandwidth, and the pump signalintensity, one can design a PBG structure to yield a desired outputsignal having a frequency different from the pump signal.

As discussed, both forward and backward SH generation can occur. Inother words, a frequency conversion device can either transmit orreflect the output harmonic signal. Additionally, assuming that themedium is dispersionless, and the pump signal is tuned at thelow-frequency band-edge transmission resonance, then the SH frequency isfound well away from the second-order band edge: it is tuned in themiddle of the pass band, as indicated in FIG. 7. In order to properlytune the SH signal frequency near the band edge, material dispersion isintroduced. This causes changes in the band structure. Specifically, allhigher-order gaps tend to move down in frequency, causing the SH signalto be tuned closer to the low-frequency, second-order band edge, wherethe electromagnetic density of states is largest.

From a calculational standpoint, varying the amount of dispersion isstrightforward to undertake. From a fabrication standpoint, obtainingthe same conditions can be more difficult. However, the inventors findthat the band structure and its features are strongly influenced by (a)the number of periods, (b) layer thickness, and (c) material dispersion.For example, increasing (or decreasing) the number of layers sharpensthe band edges, and increases (or decreases) the number of transmissionresonances between gaps, causing an effective shift of each resonance.Changing layer thickness away from the quarter- or half-wave conditions(in units of λ₀) can also cause frequency shifts in the location of theband gaps and transmission resonances. A structure with the desiredproperties can be realized when these frequency shifts are coupled withmaterial dispersion.

In order to find the optimal parameters for SH generation, i.e., tuningwith respect to the band edge, the index of refraction of the high-indexlayer is varied from n₂(2Ω)=1.42857 to n₂(2Ω)=1.65. The higher-indexvalue corresponds to SH generation just inside the second-order gap,where its suppression is expected. For intermediate values of the index,SH generation also occurs at frequencies where the density of modes is amaximum. The degree of dispersion assumed is typical of the degree ofdispersion found in both dielectric or semiconductor materials, 5-10% inthis case.

Recall that FIG. 4 shows the group index, defined as N_(g)=cdk/dω, for apreferred PBG sample, similar to that shown in FIG. 2. Note that themaximum group index is also a sensitive function of δn, the indexmodulation depth, and the number of periods. The maximum value of thegroup index for this mixed half-quarter-wave structure is similar inmagnitude to that of a quarter-wave, 20-period structure with the sameindex modulation depth. In this case, n₂ (Ω)=1.42857, and n₂(2Ω)=1.519.Note also that the magnitude of this function is largest near the high-and low-frequency band edges.

-   -   b. Picosecond Input Pulse Model

In this example, the pump pulse frequency is chosen to correspond to thelow-frequency band edge, where the transmission resonance isapproximately unity and the group index is a maximum (Ω=0.591 in FIG.2). A high pump index implies that a dramatic increase in the fieldintensities inside the structure occurs at that frequency. This isimportant, since SH gain is nonlinear in the field, as Eq. (8) suggests.By choosing the index of refraction such that n₂ (2Ω)=1.519, the SHfrequency coincides with the second density of modes maximum on thelow-frequency side of the second-order band gap (see FIG. 4). Here, thetotal-energy output from the PBG device with respect to theindex-matched bulk, which includes forward and backward SH generation,varies from one order of magnitude for a pump pulse only 60 opticalcycles in duration (1/e width of the intensity envelope is about 200femto-seconds (fs) if λ₀=1 μm), to approximately 500 times for pulsesabout 1 ps long. For sub-picosecond pulses, the enhancement is reduceddue to the broad frequency content of the pulse.

SH generation is not at a maximum when the SH signal is tuned at thedensity of the mode maximum, because the fields do not have the rightphase for this to occur. As an example, using the known matrix transfermethod, it can be found that the phase of the transmitted, plane-wavefield undergoes a π phase shift across the gap, and a phase shift of 2πbetween consecutive resonances on the same side of any gap. Therefore,the number of periods chosen can have an impact on the overall phase ofthe SH field inside the structure. For short pulses, the circumstancesare much more complicated, because of their broadband frequency makeup.

Here we present a more general effective index theory, which isdescribed in “Dispersive properties of one-dimensional photonic band gapstructures: application to quadratic nonlinear interaction”, by M.Centini et al, which can be used for a more systematic way on how toarrive at the structure's desired properties, namely, a structure wherelarge density of modes and phase matching conditions are obtained. Usingthe matrix transfer method described in the book by Fowles, for example,we define a general transmission function for any structure as follows:t≡x+iy≡{square root}{square root over (T)}e ^(iφ) _(t)   (9)where {square root}{square root over (T)} is the transmission amplitude,φ_(i)=tan⁻¹(y/x)±mπ is the total phase accumulated as light traversesthe medium, and m is an integer number. In analogy with the propagationin a homogeneous medium, we can express the total phase associated withthe transmitted field asφ_(i) =k(ω)D=(ω/c)n _(eff)(ω)D,  (10)where k(ω) is the effective wave vector; and n_(eff) is the effectiverefractive index that we attribute to the layered structure whosephysical length is D. As described in M. Centini et al, we obtain thefollowing expression of the effective index of refraction:{circumflex over (n)} _(eff)=(c/ωD)[φ_(i)−(i/2)ln(x ² +y ²)]  (11)Eq.(11) suggests that at resonance, where T=x²+y²=1, the imaginary partof the index is identically zero. We can also define the effective indexas the ratio between the speed of light in vacuum and the effectivephase velocity of the wave in the medium. We have $\begin{matrix}{{\hat{k}(\omega)} = {\frac{\omega}{c}{{\hat{n}}_{eff}(\omega)}}} & (12)\end{matrix}$

This is the effective dispersion relation of the finite structure. Forperiodic structures, the phase matching conditions are automaticallyfulfilled if the fields are tuned at the right resonance peaks of thetransmission spectrum. Using the formalism introduced in J. M.Bendickson et el, Phys. Rev E 53, 4107 (1996), the expression for theeffective index for the N-periods finite structure can be recast asfollows: $\begin{matrix}{{{\hat{n}}_{eff} = {\frac{c}{\omega\quad{Na}}\left\{ {{\tan^{- 1}\left\lbrack {{\mathcal{z}}\quad{\tan\left( {N\quad\beta} \right)}{\cot(\beta)}} \right\rbrack} + {{{Int}\left\lbrack {\frac{N\quad\beta}{\pi} + \frac{1}{2}} \right\rbrack}\pi}} \right\}}},} & (13)\end{matrix}$where β is the Bloch's phase for an infinite structure having the sameidentical unit cell as the finite structure in question. Eq.(13)contains additional information regarding the location of the resonanceswhere phase-matching for a general multi-wave mixing (MWM) process canoccur. For illustration purposes, we consider the usual 20-period, mixedhalf-wave/eighth-wave periodic structure. We choose this arrangementbecause it allows easy tuning of all the fields near their respectiveband edges, thus allowing us to simultaneously access a high density ofmodes for all fields. We note that this arrangement is not unique, inthat higher or lower order band edges can be combined to yield the phasematching conditions, within the context of the effective index approachoutlined in M. Centini et al. For simplicity, we consider the secondharmonic generation process and assume a pump field is tuned atfrequency ω; the interaction via a Ω⁽²⁾ process then generates adown-converted signal at a third frequency ω₂=2ω. First, we tune thefield of frequency ω at the first resonance near the first order bandedge; this insures the highest density of modes possible. We have:$\begin{matrix}{\beta_{1} = {\frac{\pi}{N}\left( {N - 1} \right)}} & (14)\end{matrix}$

This expression fixes the phase of the first field. Since we areinterested in second harmonic generation, we need to impose theconditions on the third generated field, namely:K ₂(ω₂)−K ₁(ω₁)=0  (15)Expression (15) fixes the condition through which exact phase matching,and hence high conversion efficiency, can be obtained. We have ensured ahigh density of modes by tuning near the band edge. Substituting (14)and (15) into (13), we obtain $\begin{matrix}{\beta_{2} = {\frac{\pi}{N}\left( {{2N} - 2} \right)}} & (16)\end{matrix}$which is the value of the Bloch's phase that will correspond to thefield at frequency 2ω. This means that phase-matching conditions will besatisfied for this structure if the thickness of the layers are combinedwith material dispersion such that the pump field is tuned to the firstresonance away from the low frequency band edge of the first order gap(N−1). Then the second harmonic field must be tuned to the secondresonance (2N−2) near its band edge. The procedure outline can berepeated for a number of fields, for either up or down-conversion, andthe results agree well with the numerical model outlined above.

FIG. 8 shows the calculated SH energy output for a 1 ps pulse, as afunction of n₂(2Ω), i.e., dispersion. The maximum energy output occurswhen n₂(2Ω)=1.519, which corresponds to the second transmission or groupindex maximum, a value that satisfies the phase-matching conditionsestablished by Eqs.(15-16) above. The band structure for n2=1.519 isillustrated in FIG. 4. Evidence of the curvature of the band structurenear the band edge is rather weak away from the second transmissionresonance. These results demonstrate that the dipole distribution isalso an important factor. In this case, the SH field is generated insidethe structure from a continuous distribution of nonlinear dipoles; thenonlinearity is in both the high and low index layers. This dipoledistribution determines the form of the propagating eigenmode, and themanner in which the generated signal leaves the structure. Therefore, aswould be apparent to one of skill in the art based on the presentdescription, one can likely find a nonlinear dipole distribution thatwill maximize or further improve SH conversion efficiency.

The above calculations also highlight the importance of pulse width.Pulses whose spectral widths are larger than the band-edge transmissionresonance tend to couple poorly with the structure. This situation leadsto dispersive propagation, and to only slightly enhanced fieldintensities inside the PBG structure. On the other hand, a pulse whosefrequency band-width is smaller than the band-edge resonance bandwidthhas fewer frequency components, experiences little or no dispersion, andallows the field to build up inside the structure by about one order ofmagnitude or more with respect to its free space or bulk values, wherethe field amplitude is in general proportional to E_(free)/n.

For example, FIG. 9 plots the pump-field intensity inside the structure,at the instant the peak of the 1-ps pulse reaches the structure. As thepulse slows down dramatically, the maximum field intensity is amplifiedby more than one order of magnitude (compared to its peak value outsidethe structure) by linear interference effects of backward- andforward-traveling components. FIG. 10, on the other hand, represents theSH field intensity quasistanding-wave pattern at the same instant intime as FIG. 9. Both eigenmodes overlap to a large extent inside thehigh index layers, and the fields propagate in this configuration forthe entire duration of the pump pulse. This mode overlap, combined withthe dramatic group velocity reduction for both fields, allows efficientenergy exchange between the pump and the SH signal.

FIG. 11 shows the total-energy output (forward and backward included) asa function of incident pulse width, expressed in optical cycles, for a20-period, 12-μm-thick device (solid line), and a 12-μm bulk samplecoated with anti-reflection layers at both ends to minimize pumpreflections (dotted line). Low input field intensities are consideredthat yield conversion efficiencies on the order of 10⁻¹², although thistrend persists as long as pump depletion is not significant. Forclarity, the abscissa is plotted on a logarithmic scale. FIG. 11 showsthat the total-energy output (and therefore power output) becomes about500 times greater for the PBG sample than for index-matched bulkmaterial when an input pulse width approaches 300 optical cycles, orabout 1 ps. The results indicate that at these length scales the energyoutput for the bulk sample increases linearly with incident pulse width.Thus, this figure clearly demonstrates that suitable output energies canbe obtained from the PBG devices of the present invention whencontinuous wave input pulses are applied.

In contrast, an early exponential increase characterizes energy growthin the PBG case, giving way to linear growth only when pulse widthapproaches 1 ps. This implies that the pump field eigenmode intensity(and hence SH gain) increases rapidly with pulse width, saturating whena quasi-monochromatic limit is reached, in this case, when pulsefrequency bandwidth is somewhat less than band-edge resonance bandwidth.

Also, both the amplitude and the width of the generated SH pulsesincrease with increasing incident pulse width. FIG. 12 shows the SHfield propagating away from the structure. While the pump was incidentfrom the left, note that the structure radiates significantly in bothdirections, and that the SH pulses generated have the same width asincident pump pulses. It would be difficult to predict this overallbehavior a priori, especially in the absence of analytical results inthis regime. Further, tuning the pump away from the band edge, tuning tothe high-frequency band edge, or modifying the nonlinear dipoledistribution can significantly alter the pattern of FIG. 12.

FIG. 13 is a plot of the conversion efficiency vs. peak field intensityin Gaussian units, for a pulse of 1 ps duration, where |E|² of 10⁹ inthese units corresponds to roughly 10 GW/cm² in free space. Thefree-space value of the energy flow is to be distinguished from energyflow inside the structure. Here, efficiency is defined as the ratiobetween the final total SH energy and the total initial pump energy.This ratio is also representative of the ratio between the correspondingpeak field intensities, respectively. FIG. 13 indicates that for thissimple PBG structure only 12 μm in length, a conversion efficiency oforder 10⁻² can be achieved with pump intensity of 10 GW/cm², yielding aSH signal intensity of approximately 0.1 GW/cm². This is quiteremarkable, considering that the PBG structure is only a few micrometersin length, only a single pump pass occurs, and that a very modest valueof Ω⁽²⁾≈0.1 pm/V is used. Note that a Ω⁽²⁾ value of 0.1 pm/V is aconservative value. Clearly, materials chosen with even higher Ω⁽²⁾values can be incorporated into the PBG structure of the presentinvention, resulting in conversion efficiencies approaching 10⁻1.Considering the extremely compact nature of the PBG device of thepresent invention, and that the pump traverses the sample only once, thegain-to-device length ratio undergoes several orders of magnitudeimprovement over current state of the art devices.

Such large enhancements with respect to phase-matched up-conversion canbe explained as follows. According to Fermi's golden rule, the powerradiating from an oscillating dipole is given by P(ω)=ρ(ω)|E(ψ)|², whereρ(ω) is the density of modes and |E(ω)|² is the eigenmode intensity. Theaverage energy output can be obtained by multiplying the power output byτ, the interaction time. As pointed out above, all these quantitiesincrease by nearly one order of magnitude for the PBG structure. Infact, since |E(ω)|² and r are both proportional to ρ(ω), then the totalenergy emitted is generally proportional to ρ(ω)₃. Hence the significantincrease in the total energy output that is shown in FIG. 11.

Higher conversion efficiencies can readily be achieved by increasingpump power, or, as mentioned earlier, by increasing the length of thestructure by only modest amounts. For example, calculations show that byincreasing the total number of periods to 30, thus increasing the lengthof the device by 50%, the SH output energy (and power level) increasesby a factor of 5 for a 1 ps pulse, enhancing the conversion efficiencyby the same factor. This occurs because the maximum group indexincreases approximately as N², where N is the number of periods. Thefield eigenmode intensity is also proportional to N², thus enhancingenergy output in a nonlinear fashion with respect to device length.

Calculations also indicate that in the linear, undepleted-pump regime,the conversion efficiency is proportional to the free-space peak fieldvalue, as illustrated in FIG. 13. Here, any small deviation in theactual Ω⁽²⁾ value, tuning with respect to the band edge, and input pulsewidth can significantly affect comparison with experimental results. Forthis reason, the model presented above is of great value in order todetermine the overall behavior of a PBG structure, and it can be used inthe determination of Ω⁽²⁾. Therefore, exercising reasonable care in thedesign process of a PBG device based on the present invention canproduce a very efficient SH generator, provided absorption at the SHwavelength is minimized. Note that a similar model can also be used todesign an efficient third harmonic generator, and the like.

In another embodiment of the present invention, a structure comprises aseries of alternating layers, where nv(Ω,2Ω)=1 and n₂(Ω,2Ω)=1.42857. Foradded simplicity, it is assumed that the material is not dispersive. Forthis example, layer thicknesses are chosen such that the width of thelow index layer is a=0.65λ₀/n₁ (the low index layer is now the activelayer because of the shift in localization of the field), and the widthof the high index layer is such that b=0.089λ₀/n₂. Then, tuning the pumpat the first resonance of the first-order, high-frequency band edgecauses the SH signal to be tuned at the second resonance of thesecond-order high-frequency band edge, in analogy to what wasaccomplished above for the low-frequency band edge. However, theconversion efficiency for the high frequency band edge example canincrease up to about a factor of two for a 1 ps pulse, compared to thelow-frequency band-edge conversion efficiency. Tuning the pump at thehigh-frequency band edge causes a shift of the pump field localizationin the low index layer. This shift increases the field eigenmodeintensity in that layer. Also, the width of the active layer increasesby about 30%, from 0.5λ₀ to 0.65λ₀. This combination can account for theincrease in overall nonlinear gain for a device length of approximately12 μm in length.

c. GaAs/AlAs and AlN/SiO₂ Half-Quarter-Wave Stack for SH Generation

This section describes a numerical model of a mixed half-quarter-wavestructure comprising 20 periods of GaAs/AlAs material. It was assumedthat Ω⁽²⁾≈1 pm/V for both materials, the index of refraction alternatedbetween n₁(Ω)=2.868 and n₂(Ω)=3.31, n₁(2Ω)=2.9 and n₂(2Ω)=3.35, and thatabsorption could be ignored. These indices correspond to a pumpwavelength of 3 μm, and a second-harmonic signal at 1.5 μm. For a pumpintensity of 10 GW/cm², the mixed half-quarter-wave GaAs/AlAs structureproduced conversion efficiencies on the order 10⁻²-10⁻³ for this20-period structure. The model equations described above for nonlinearSH gain (which from Eq. (6) is defined as the product Ω⁽²⁾−² _(Ω))indicate that this high conversion efficiency is due to theorder-of-magnitude increase in Ω⁽²⁾ and the order-of-magnitude decreasein the field eigenmode intensity (due to the substantial increase in theindex for GaAs). In addition, a significant increase in conversionefficiency can be achieved with increasing number of periods in the PBGstructure. These results also indicate that different materials, such asII-VI based semiconductors, would be ideal for up-converting at higherfrequencies.

As another example, we consider 30-periods of alternating SiO₂/AlNlayers, to form a quarter-wave/half-wave stack similar to that describedin the previous section. With a reference wavelength of 0.52 microns,the total length of the structure is approximately 6 microns. The AINlayers are assumed to have a nonlinear coefficient of about 10 pm/V. TheFF beam is assumed to be incident at an angle of 30° degrees withrespect to the surface of the structure. The FF frequency, at 800 nm, istuned to the first transmission resonance near the first order band gap,where the corresponding linear mode is well localized, and a highdensity of modes is achieved. The SH frequency is tuned to 400 nm, whichcorresponds to the second resonance peak near the second order gap. Wenote that we are using experimentally available data for both materials,and that aligning the resonances as prescribed can be done by varyingthe thickness of the layers, i.e., by adjusting the geometricaldispersion of the structure. The structure is exactly phase-matchedΔk_(eff)=0 in the sense of the effective index of Centini et al, and thepredictions are for increased conversion efficiency.

6. Method of Frequency Conversions—Harmonic Generation

A method for performing frequency conversion of an input photonic signalis shown in FIG. 14. The input photonic signal has an input photonicsignal frequency and an input photonic signal bandwidth. In step 1402,the frequency of the input photonic signal is selected so as tocorrespond to a second signal at a desired harmonic frequency. Inaddition, the type of input signal (e.g., continuous wave or pulsedoperation) should also be considered. Next a device is provided in step1404, where the device comprises an arrangement of material layers thatexhibits a photonic bandgap structure. Various types of material layerarrangements are discussed above. The specific type of arrangement (andhence the type of frequency conversion to be performed) depends uponfactors that include, but are not limited to: (1) the absorption andtransmission properties of the materials selected; (2) the indices ofrefraction of the materials forming the structure, which affects suchparameters as the index discontinuity; (3) the thicknesses of thematerial layers; and (4) the number of periods of alternating layers.The combination of parameters results in a PBG structure that preferablyexhibits a transmission band edge corresponding to the input photonicsignal frequency. Finally, the input photonic signal is delivered intothe device in order to generate a second photonic signal at an harmonicfrequency of the pump signal. An interaction of the input photonicsignal with the arrangement of layers generates the second photonicsignal at a second frequency, where the second frequency is differentthan the first frequency. It will be apparent to one of skill in the artto use this method to perform such frequency conversion techniques as,for example, harmonic generation and optical parametric oscillation.

7. Frequency Down-Conversion

Another aspect of the present invention is directed to photonic signalfrequency down-conversion using a PBG structure similar to the abovedescribed structures, as is now described. FIG. 15 is a block diagram ofan embodiment of a system 1500 for frequency down-converting photonicsignals. System 1500 includes a first pump light source 1502, such as alaser, for generating a first coherent photonic beam or signal 1504 at afirst frequency ω₃. A second pump light source 1506, such as a laser,generates a second coherent photonic beam or signal 1508 at a secondfrequency ω₂. In one embodiment, each of pump light sources 1502 and1504 generates pulsed laser light. In another embodiment, each of pumplight sources 1502 and 1504 generates Continuous Wave laser light. Instill another embodiment, pump light source 1502 generates pulsed laserlight while pump light source 1504 generates Continuous Wave laserlight.

System 1500 includes a PBG device 1520 constructed and arranged inaccordance with the present invention to frequency down-convert photonicsignals incident on the device. PBG device 1520 includes an input face1522 for receiving incident photonic signals 1504 and 1508, and anoutput face 1524. PBG device frequency down-converts photonic signals1504 and 1508 to generate a coherent, frequency down-converted outputphotonic signal 1530 having a frequency ω₁=ω₃−ω₂, which is partiallytransmitted and exits output face 1524, and is partially reflected andexits from face 1522.

Down-converted output photonic signal 1530 results from a photonicfrequency mixing process, within the PBG structure of PBG device 1520,between first and second photonic signals 1504 and 1508 incident on thePBG structure. Assuming the first and second photonic signals 1504 and1508 have respective center frequencies ω₃ and ω₂, the frequency mixingprocess within PBG device 1520 generates the following coherent,frequency conversion products (also referred to as mixing products): thethird photonic signal (e.g., photonic signal 1530) at differencefrequency ω₁=ω₃−ω₂; and a fourth photonic signal at a sum frequencyω₄=ω₃+ω₂. The third photonic signal at down-converted differencefrequency ω₁, represented by output photonic signal 1530, is of interestin the down-conversion process, while the sum of the frequencies is ofinterest in the up-conversion process, which we will discuss later.

FIG. 16 is an embodiment of PBG device 1500 capable of frequencydown-converting photonic signals incident on the device, in theexemplary manner described above. PBG device 1500 includes a firstmaterial layer 1602 and a second material layer 1604 together forming asingle period 1606. For a reference wavelength λ₀, layers 1602 and 1604have respective refractive indices n₁ and n₂, and respective thicknessesa=λ₀/(8n₁) and b=λ₀/(2n₂), to form respective eighth-wave and half-wavelayers in single period 1606. Single period 1606 is repeated to form amixed half-eighth wave multilayer stack for wavelength λ₀. Therefore,PBG device 1500 is also referred to as a mixed half-eighth wavemultilayer stack 1500.

FIG. 17A is a transmission function 1700 of the example half-eighth wavemultilayer stack of FIGS. 15 and 16. Transmission function 1700 is aplot of the transmission coefficient of PBG device 1500 versus thenormalized or scaled frequency Ω=ω/ω₀, where ω₀=2 πc/λ₀. The arrangementof layers in device 1500 causes transmission function 1700 to exhibitthe following three harmonically related bandgaps: a high order or highfrequency bandgap 1705; an intermediate frequency bandgap 1710; and alow order or low frequency bandgap 1715.

High frequency bandgap 1705 includes a low order bandgap edge 1720, anda transmission resonance 1722 corresponding to a photonic frequencyω₃=3ω. Intermediate bandgap 1710 includes a low order band edge 1725corresponding to a photonic frequency ω₂=2ω Low order bandgap 1715includes a low order band edge 1730 corresponding to a photonicfrequency of ω₁=ω. With reference again to FIG. 15, when input photonicsignals 1504 and 1508 have respective frequencies ω₃=3ω and ω₂=2ω, thatis, frequencies tuned at or near transmission resonances 1722 and 1725,PBG device 1500 generates down-converted output photonic signal 1530 atdifference frequency: ω₁=ω₃−ω₂, where frequency ω₁ is tuned to low orderband edge 1730.

Tuning the ω₁, ω₂, and ω₃ fields as depicted in FIG. 17A meets phasematching conditions conducive to photonic signal down-conversion in thepresent invention. However, these tuning conditions are not unique inthat one may tune the fields in a slightly different way to satisfy thephase matching conditions similar to those outlined above inEqs.(14-16). The precise nature of the phase matching conditions whenmore than two fields are present are outlined in the manuscript entitled“Efficient nonlinear infrared generation in one dimensional photonicband gap materials” by M. Centini et al, submitted for publication toOptics Communications. With reference to FIG. 17B, the phase matchingconditions can also be satisfied by arranging the fields in a slightlydifferent way. Generally, this kind of flexibility cannot be exploitedin ordinary nonlinear frequency conversion materials, and is one of themore attractive features of our invention.

The criteria resulting in a structure suitable for frequencydown-conversion in PBG device 1500 are now further described withreference to an example three-wave (that is, three-photonic signal)frequency mixing process. The example three-wave mixing processincludes:

-   -   a first pumping field (for example, photonic signal 1504) tuned        to frequency ω₃=3ω, corresponding to a wavelength (λ₃) of 1        micron; and    -   a second pumping field (for example, photonic signal 1508) tuned        to frequency ω₂=2ω, corresponding to a wavelength (λ₂) of 1.5        microns.

These first and second pumping fields interact such that theirrespective frequencies ω₃(λ₃=1 micron) and ω₂(λ₂=1.5 micron) mix togenerate sum and difference frequencies, as described above. Forfrequency down-conversion, conditions are established such that a field(that is, a photonic signal) is generated at the difference frequencyω₁=(ω₃−ω₂, having a corresponding wavelength of 3 microns. The choice ofpumping fields at wavelengths of 1 and 1.5 microns is convenient forillustrative purposes (but is not necessary) because frequency mixingbetween the 1 and 1.5 micron fields results in a down-converted fieldhaving a wavelength (λ₁) of 3 microns.

The particular structure of PBG 1500, namely, the mixed half-eighth wavemultilayer stack significantly enhances an interaction between the threewaves (i.e., fields) at frequencies ω₃=3ω(λ₃=1 micron), ω₂=2ω(λ₂=1.5micron) and ω₁=ω(λ₁=3 micron). With reference again to FIG. 17A, thethree fields ω₁, ω₂, and ω₃ are depicted as being tuned to respect bandedge features 1722, 1725, and 1730. With the three fields tuned asdepicted in FIG. 17A, a high density of modes is achieved. The fieldsω₁, ω₂, and ω₃ have corresponding frequencies ω₁=ω, ω₂=2ω, and ω₃=ω₂+ω₁(in this case, ω₃=3ω) with k₁≡k=ω/c, k₂=2ω/c, and k₃=3/c. Following theformalism of Eqs.(14-16) above, tuning the fields as in FIG. 17A leadsto the following conditions on the wave vectors: $\begin{matrix}{\beta_{1} = {\frac{\pi}{N}\left( {N - 1} \right)}} & (17)\end{matrix}$

Then, we tune the field ω₂ at the first resonance near the second orderband edge, once again securing a high density of modes; we have:$\begin{matrix}{\beta_{2} = {\frac{\pi}{N}\left( {{2N} - 1} \right)}} & (18)\end{matrix}$

We now impose the phase-matching condition for the three-wave mixingprocess, namely:K ₃(ω₃)−K ₂(ω₂)−K ₁(ω₁)=0  (19)

Substituting as before, we have the condition on the third wave, namely:$\begin{matrix}{\beta_{3} = {\frac{\pi}{N}\left( {{3N} - 2} \right)}} & (20)\end{matrix}$which is the value of the Bloch's phase that will correspond to thefield at frequency ω₃. This means that phase-matching conditions will besatisfied for this structure if the thickness of the layers are combinedwith material dispersion such that the first pump field is tuned to thesecond resonance away from the low frequency band edge of the thirdorder gap (3N−2).

In order to illustrate the flexibility of the structure, the phasematching condition for the three-wave mixing process can also beobtained as outlined on FIG. 17B. In that case, the wave vectors inEqs.(17), (18) and (20) have the following factors, respectively: (N−1),(2N−2), and (3N−3), conditions which yield nearly identical results asthe conditions of FIG. 17A.

FIG. 18 includes a plot of an example energy function 1802 for the ω₃pump field, a plot of an example energy function 1804 for the ω₂ pumpfield, and a plot of an example energy function 1806 for thedown-converted field ω₁, when PBG device 1500 is pumped with shortphotonic pulses (having approximately 0.5 picosecond durations, forexample) of the ω₃ and ω₂ pump fields. Each of the energy functions is aplot of total integrated energy vs. time for each associated field, asthe ω₃ and ω₂ field pump pulses sweep through PBG device 1500. A steadystate is reached as soon as the pump pulses leave the PBG structure.

With reference to FIG. 18, the down-converted field ω₁ and pump field ω₂experience effective gain during the mixing interaction of the threewaves. However, the ω₃ pump field is nearly 50% depleted. For thisspecific example, the layered PBG structure 1500 yields:

-   -   (1) conversion efficiencies approximately 500 times better        compared to the efficiency of bulk;    -   (2) a depletion of nearly 50% of the 1-micron pump;    -   (3) a conversion rate of 20% for the 3-micron field; and    -   (4) a gain of 30% for the 1.5-micron signal.

Considering that incident pump fields are of order 10⁷ V/m (100 MW/cm²flux), and that Ω⁽²⁾ is a modest 100 pm/V, the above described effectsreveal a remarkable photonic frequency down-conversion capability.Furthermore, SH generation in bulk materials is generally proportionalto L², where L is the length of the PBG device 1500. In the case of PBGstructures, phase matching and band edge effects yield a SH intensityproportional to L⁶. A length L=NΛ, where N is the number of periods andΛis the thickness of the unit cell (period), conversion efficiencies canbe improved even more dramatically by adding only a few periods.

The formalisms and examples described above for the down-conversionprocess also apply to the frequency up-conversion process. For example,the above described model can be inverted as follows: instead ofinjecting an ω₃ and ω₂ field, we simply inject an ω₁ field, which has alower frequency. Then using the conditions outlined in FIG. 17B, theunusual, unprecedented set of circumstances arise: there two competingprocesses that take place. First, second harmonic generation occurs, andour structure provides exact phase matching for that process. Second,the resulting fields mix to generate the third frequency, or ω₃, whichin this case corresponds to third harmonic generation. Remarkably, thisprocess is also phase matched. Therefore, we have two competingprocesses that are simultaneously phase matched, which means thatdepending on the precise nature of the coupling coefficients, i.e.,number of layers, tuning, layer thickness, etc., we may selectivelyenhance either second or third harmonic generation, once againhighlighting the great flexibility afforded by photonic band gapstructures. We note that under ordinary conditions, phase matching athree-wave mixing process in the manner described above and asillustrated in our FIG. 17B is difficult to achieve in ordinarymaterials.

Current technology allows the construction of microscale, layeredstructures. It is expected that such construction techniques as appliedto the above described PBG photonic frequency down-converting structureswill have significant impact in dual use applications. The 3-5 micronwavelength range is important for remote sensing, laser ranging,blinding of enemy seekers, and detection of biological and chemicalcomponents. PBG down-conversion devices as described above can reducethe volume of an active portion of a portable sensing unit by up toseven orders of magnitude (to produce, for example, a wafer having 10 cmsides, and a length of 10-40 microns), thus making space-basedapplications possible. Current terrestrial wind measurement projectscarried out by NASA investigators, for example, use high power IRsources whose size prevents their effective use in satellite platforms.The present invention advantageously overcomes such a size limitation.

8. Conclusions

While various embodiments of the present invention have been describedabove, it should be understood that they have been presented by way ofexample only, and not limitation. Thus, the breadth and scope of thepresent invention should not be limited by any of the above-describedexemplary embodiments, but should be defined only in accordance with thefollowing claims and their equivalents. Additionally, all articles andpatent documents mentioned above are incorporated by reference herein.

1. A device for frequency down-converting a photonic signal incidentupon the device, comprising: a plurality of first material layers; and aplurality of second material layers, the first and second materiallayers being adapted to receive first and second photonic signalsincident upon the device and having respective first and secondfrequencies, the first and second material layers being arranged suchthat the device exhibits a photonic band gap structure, wherein thephotonic band gap structure exhibits first and second transmission bandedges respectively corresponding to the first and second frequencies,and wherein an interaction of the first and second photonic signals withthe arrangement of layers causes a mixing process to generate a thirdphotonic signal having a third frequency that is less than the first andsecond frequencies.
 2. The device of claim 1, wherein said first andsecond material layers are arranged in a periodically alternating mannersuch that the arrangement formed therefrom exhibits said photonic bandgap structure.
 3. The device of claim 1, wherein said first materiallayer has a first index of refraction and said second material layer hasa second index of refraction, said first index of refraction and saidsecond index of refraction being different.
 4. The device of claim 1,wherein said first material layer has a first thickness and said secondmaterial layer has a second thickness, said first thickness and saidsecond thickness being different.
 5. The device of claim 1, wherein saidphotonic band gap structure also exhibits a set of transmissionresonances near the third order band gap, and wherein said thirdfrequency is tuned such that phase matching conditions are satisfied toenhance the generation of the third frequency.
 6. The device of claim 1,wherein each of said first and second input photonic signals is oneof 1) a continuous wave photonic signal generated by a continuous wavelaser source, and 2) a pulsed photonic signal generated by a pulsedlaser source.
 7. The device of claim 1, wherein said arrangement oflayers forms a mixed half-eighth wave multilayer stack, and said secondfrequency is 2ω, and the input photonic signal frequency has frequency3ω.
 8. The device of claim 1, wherein said arrangement of layers forms amixed half-eighth wave structure.
 9. The device according to claim 1,wherein said first and second material layers respectively comprise GaAsand AlAs semiconductor layers, said first and second layers being formedon an appropriate substrate.
 10. The device according to claim 1,wherein said first and second material layers respectively comprise AlNand SiO₂ layers, said first and second layers being formed on aappropriate substrate.
 11. The device of claim 1, wherein a length ofthe device is between a few hundred nanometers and a few thousandmicrons.
 12. A method of frequency down-converting a photonic signalincident on a device, the device including a plurality of first materiallayers and a plurality of second material layers, the first and secondmaterial layers being arranged such that the device exhibits a photonicband gap structure, wherein the photonic band gap structure exhibitsfirst and second transmission band edges, the method comprising thesteps of: applying first and second photonic signals to the first andsecond material layers, the first and second photonic signals havingrespective first and second frequencies corresponding to the first andsecond transmission band edges, wherein an interaction of the first andsecond photonic signals with the arrangement of layers causes a mixingprocess to generate a third photonic signal having a third frequencythat is less than the first and second frequencies.
 13. The method ofclaim 12, further comprising the step of mixing the first and secondfrequencies such that the third frequency is the difference between thefirst and second frequencies.
 14. The method of claim 12, wherein themixing step generates the third frequency such that the third frequencyis tuned to a third transmission resonance associated with a third bandgap edge.
 15. The method of claim 12, wherein a number of input beamsmay be injected to a plurality of first and second layers, such thatphase matching conditions are satisfied.
 16. A device for frequencyup-converting a photonic signal incident upon the device, comprising: aplurality of first material layers; and a plurality of second materiallayers, the first and second material layers being adapted to receive afirst photonic signal incident upon the device and having a firstfrequency, the first and second material layers being arranged such thatthe device exhibits a photonic band gap structure, wherein the photonicband gap structure exhibits first and second transmission band edgesrespectively corresponding to the first and second frequencies, andwherein an interaction of the first photonic signal may generate asecond photonic signal with a frequency near the second band edge, andsuch that the arrangement of layers causes a further mixing process togenerate a third photonic signal having a third frequency that is morethan the first and second frequencies.
 17. The device of claim 16,wherein said first and second material layers are arranged in aperiodically alternating manner such that the arrangement formedtherefrom exhibits said photonic band gap structure.
 18. The device ofclaim 16, wherein said first material layer has a first index ofrefraction and said second material layer has a second index ofrefraction, said first index of refraction and said second index ofrefraction being different from one another.
 19. The device of claim 16,wherein said first material layer has a first thickness and said secondmaterial layer has a second thickness, said first thickness and saidsecond thickness being different from one another.
 20. The device ofclaim 16, wherein said photonic band gap structure also exhibits a thirdtransmission resonance at a third order band gap, and wherein said thirdfrequency is tuned to said third transmission resonance.
 21. The deviceof claim 16, wherein the input photonic signal is one of 1) a continuouswave photonic signal generated by a continuous wave laser source, and 2)a pulsed photonic signal generated by a pulsed laser source.
 22. Thedevice of claim 16, wherein said arrangement of layers forms a mixedhalf-eighth wave multilayer stack, and said second frequency is a secondharmonic of the input photonic signal frequency, and said third photonicsignal is a third harmonic of the input photonic signal frequency.
 23. Amethod of frequency up-converting a photonic signal incident on adevice, the device including a plurality of first material layers and aplurality of second material layers, the first and second materiallayers being arranged such that the device exhibits a photonic band gapstructure, wherein the photonic band gap structure exhibits first andsecond transmission band edges, the method comprising the steps of:applying a first photonic signal to the first and second materiallayers, generating a second photonic signals having a second frequencycorresponding to the second transmission band edge, wherein a subsequentinteraction of the first and second photonic signals with thearrangement of layers causes a mixing process to generate a thirdphotonic signal having a third frequency that is more than the first andsecond frequencies.
 24. The method of claim 23, wherein a first andsecond photonic signal are injected inside the plurality of layers. 25.The method of claim 23, further comprising the step of mixing the firstand second frequencies such that the third frequency is the sum of thefirst and second frequencies.
 26. The method of claim 23, wherein themixing step generates the third frequency such that the third frequencyis tuned to a third transmission resonance associated with a third bandgap edge.
 27. The method of claim 23, wherein a number of input beamsmay be injected to a plurality of first and second layers, such thatphase matching conditions are satisfied.